Feynman's Rest of the Universe

When we solve a quantum-mechanical problem, what we really do is
divide the universe into two parts - the system in which we are interested
and the rest of the universe. We then usually act as if the system in
which we are interested comprised the entire universe. To motivate the use
of density matrices, let us see what happens when we include the part of
the universe outside the system.

Feynamn then used the density matrices and Wigner functions to illustrate
his rest of the universe. However, he used only one oscillator to
illustrate what he said about the rest of the universe. Yes! The harmonic
oscillator is the basic tool to illustrate the Wigner function. But how
could he explain two different worlds with one oscillator?

It appears that the best way to understand what Feynman said above is
to use two coupled harmonic oscillators. One of them is in the world in
which we do physics, and the other is for the rest of the universe. It
is straight-forward to construct the Hamiltonian for the system of
two coupled oscillators. It is not difficult to construct wave funtions,
density matrices, as well as Wigner functions for this two-oscillator
system. Then there is a well-established procedure for constructing the
density matrix and Wigner function when one of the two oscillators is
not obseved: to sum over or integrate out the un-observed variable.
The net result is the increase in entropy through the density matrix,
and an addition of statistical uncertainty through the Wigner function.

However, it was difficult for me and my co-authors to publish the
paper saying this. The reason is very simple. The referees kept saying
the paper is too trivial, but the truth was that they did not like my
research group becoming more famous than they are by talking about
Feynman. This is the Herod complex
shared by all physicists.

It took us eight years to publish, and it appeared finally
in the American Journal of Physics in 1999. The final referee said
the paper is right and appropriate for the journal, but he could not
recommend publication. He did not explain why. The editor over-ruled
the referee's Herod complex after our appeal.

Click here for the
article using two oscillators to illustrate Feynman's rest of the universe.
One oscillator for the world in which we do physics, and the other for
the rest of the universe.

Two-mode Squeezed States

P. A. M. Dirac was quite fond of harmonic oscillators and the
Lorentz group. He attempted to construct a representation of using two
oscillators, and ended up with the O(3,2) deSitter group applicable to
five dimensional space with three space and two time coordinates.
Here is his paper.

In this paper, he starts with the step-up and step-down operators for two
oscillators attempted to construct a Lie algebra with a closed set of
commutators. He then ended up with ten generators given
in this page.

This big O(3,2) group has many small O(2,1) subgroups. Every O(2,1) subgroup
is the Lorentz group applicable to two space and one time dimensions. One
of them later became the basic language for two-photon coherent states
commonly called "two-mode squeezed states." While both photons can be observed
in laboratories, we can pretend not to see it. In this way, we can see
clearly what Feynman was talking about when he mentioned the rest of the
universe.

With this preparation, we can become ambitious enough to tackle the
problem of hidden variables in quantum mechanics. The question is whether
they are hidden in Feynman's rest of the universe.

Time Separation hidden in Feynman's Rest of the Universe.

They could have talked the Lorentz-boosted Bohr radius, but they
did not. The time separation was a hidden variable to them.

Niels Bohr (1885-1962) was born six years after Einstein. Whenever Bohr
mentions the word "space" in his philosophical writings, he adds "time."
This is how much Bohr respected Einstein. Einstein also had a
great respect for Bohr. At the same time, Einstein had his objections to
the Copenhagen interpretation of quantum mechanics. This aspect is
well known and is still actively debated by the present generation of
physicists.

Niels Bohr is also well known for his Bohr radius. To him, the proton
was sitting at the center of the universe, even though he knew the existence
of the time variable associated with the space. Einstein perhaps thought
about the hydrogen atom in a moving frame, but he never said anything
about it. His Lorentz transformations are strictly for point particles.

Before saying there was something wrong with them, let us look at Thomas
Jefferson. He said "All men are created equal." But he forgot to mention
"women" there, and he was not able to practice his concept of equality
with some of the the people who served him at his Virginia mansion called
Monticello.
Jefferson's freedom remained as a hidden variable to at least one half the
world population for many years. It took many years of evolutionary
processes to see the full extent of this variable. It is quite possible
that this variable is still not completely discovered.

Let us go back to Bohr and Einstein. Physics is an experimental science.
It is still unthinkable for the hydrogen atom to move with a relativistic
speed, but the proton can be accelerated to a speed very close to that of
light. The question then is whether the proton has it own quantum
mechanics like the hydrogen atom. Yes, the quark model formulated in
1964 could tell the story. We can therefore illustrate the evolution of the
hydrogen atom has gone through with
this figure.

Needless to say, this evolution was made possible thanks to the advances
in the accelerator physics. At the proton proton speed of 0.9999999999c.
At magnitude of time separation is essentially the same as that of the
spacial separation.

In order to study this hidden variable, I spent many years with Marilyn
Noz. We published many papers, but we also had many problems with
the referees, because the time-separation variable is thoroughly hidden
to them.

The first person who recognized the existence of this variable was
Richard Feynman. Based on his talk given at the 1970 APS spring meeting,
he published his paper with his students in
Phys. Rev. D 3 , 2706 (1971)]. In this paper, Feynman
et al. state the existence of the time-separation variable.
They said they could not do anything about this variable because they do
not anything about it. I like very much their honest statement.

As for the quantum mechanics of the time variable, Paul A. M. Dirac
worried about the time-energy uncertainty relation in his paper

P. A. M. Dirac, Proc. Roy. Soc. (London) A114, 243 (1927).

He noted in this paper that the time-energy uncertainty relation is a
c-number relation, and that this lack of time-like excitations makes
it difficult to combine quantum mechanics with special relativity. In his
1945 paper:

P. A. M. Dirac, Proc. Roy. Soc. (London) A183 , 284 (1945).

Dirac uses the four-dimensional harmonic oscillator to understand
the Lorentz-covariant world. The fourth variable is the time variable.
I assume he had in mind the time-separation variable. If we combine
Dirac's 1927 paper with his 1954 paper, we end up with
this figure with a circle.

In 1949, Einstein became 70 years old, and the Reviews of Modern physics
published a special issue. This issue contains Dirac's paper:

It is then not difficult to combine the circle and the rectangle to construct
the ellipse. Does this sound like a science fiction to you? We observe this
effect whenever we turn on a particle accelerator. This is a picture of
the proton released from the accelerator.
Click here for a detailed story.

Paul A. M. Dirac was of course a great physicist. I seem to work strictly
within his framework. I could do my own research because he left some
soft spots in his papers.

Dirac's papers are poems, but have you seen figures there. I seem to
enjoy translating his poems into cartoons.

Dirac wrote many history making papers. However, Dirac seldom takes
advantage of his own papers published earlier. It is therefore profitable
to link up the papers Dirac published at different times.

Dirac had a "famous-brother-in-law" named Eugene Wigner who wrote
a paper on representations of the Poincare group in 1939. In his 1949
paper, Dirac said the task of constructing relativistic quantum mechanics
is the same as that of constructing a suitable representation of the
Poincare group.

When I talked with Dirac in 1978, he was still uncomfortable about the lack
of time-like excitations in the Lorentz-covariant world. He used to call
this space-time asymmetry. He could have learned about Wigner's O(3)-like
little group to clarify the question of space-time asymmetry.

Time Separation as a Hidden Variable: In spite of all I said above, the
time separation remains as a hidden variable to the absolute majority of
physicists even these days, especially to the orthodox Copenhageners. How do you
define the word "hidden" in physics? This concept came from the uncertainty
principle which does not allow a point in the phase space of position and
momentum.

The best way to approach this problem is to use the Wigner phase space,
where position and momentum is confined in an area not smaller than Planck's
constant. The area of uncertainty can increase if we fail to measure
other variables. In that case, the region of uncertainty expands. In order
to clarify this, we can use Feynman's rest of the universe.

The mathematics of the coupled oscillator system is exactly the same as the covariant
harmonic oscillator which leads to Dirac's form of relativistic quantum mechanics,
where one of the variable is the time-separation variable. It is not
necessaary to write down lengthy formulas. The best way is to use a cartoon. It
is even better if the cartoon is the one you have seen before. The net result is

The space-time portion of this paper is from the paper I published with Wigner
in 1990. The title of the paper is
Entropy and Lorentz Transformations. If we do not measure the
time-separation variable assuming it is hidden in Feynman's rest of the
Universe, the net result is an increase in entropy and an expansion of the
uncertainty region in the Wigner phase space.
Click here for the phase space expansion.

This figure can be used for the two-photon coherent system or two-mode squeezed
state. The labels "first photon" and "second photon" were added in my
paper with Noz entitled "Time separation as a hidden variable in the Copenhagen school
of quantum mechanics," in Advances in Quantum Theory, American Institute of
Physics Conference Series No.1327, pp 138-147 (2001).
ArXiv. As far as I can see, this
story can be traced to Dirac's 1963 paper.

The organization of this webpage can be improved. please come again.
Making a webpage is like writing a paper. You know how difficult it is to
write papers.

copyright@2009 by Y. S. Kim, unless otherwise specified.
The photos of Bohr and Einstein and of Dirac and Wigner are from the E. Segre Visual Archives
of the American Physical Society.